2 Meanness of arbitrary path super subdivision graph of P

Volume51 (2011), Pages 41–48

On the meanness of arbitrary path super subdivision of paths

A. Nagarajan

Department of Mathematics V.O. Chidambaram College Thoothukudi – 628 008, Tamil Nadu

India

[email protected]

R. Vasuki

Department of Mathematics

Dr. Sivanthi Aditanar College of Engineering Tiruchendur – 628 215, Tamil Nadu

India

[email protected] Abstract

LetG(V, E) be a graph with p vertices and q edges. For every assign- ment f : V (G) → {0, 1, 2, . . . , q}, consider an induced edge labeling f^∗:E(G) → {1, 2, . . . , q} defined by

f^∗(uv) =

 f(u)+f(v)

2 iff(u) and f(v) are of the same parity,

f(u)+f(v)+1

2 otherwise

for every edge uv ∈ E(G). If f^∗(E) = {1, 2, . . . , q}, then we say that f is a mean labeling ofG. If a graph G admits a mean labeling, then G is called a mean graph. In this paper, we study the meanness of arbitrary path super subdivision of the graphPn.

1 Introduction

LetG(V, E) be a graph with p vertices and q edges. For notation and terminology we follow [1].

The concept of mean labeling was first introduced by Somasundaram and Ponraj [7]. They studied in [3, 4, 7, 8] the meanness of many standard graphs such asPn, Cn,Kn(n ≤ 3), the ladder, the triangular snake, K1,2, K1,3, K2,n, K2+mK1, Kn^c +

2K2, Sm+K1, Cm∪ Pn (m ≥ 3, n ≥ 2), quadrilateral snake, comb, bistars B(n), Bn+1,n, Bn+2,n, the corona of ladder, subdivision of central edge ofBn,n, subdivision of the starK1,n(n ≤ 4), the friendship graph C3⁽²⁾, crownCn K1, Cn⁽²⁾, the dragon, arbitrary super subdivision of a path etc. In addition, they proved that the graphs Kn(n > 3), K1,n(n > 3), Bm,n(m > n + 2), S(K1,n)(n > 4), C3^(t)(t > 2), and the wheelWnare not mean graphs. Also some standard results are proved in [2, 5].

A vertex labeling ofG is an assignment f : V → {0, 1, 2, . . . , q}. For a vertex labelingf, the induced edge labeling f^∗ is defined by

f^∗(uv) =_f(u)+f(v)

2



for any edgeuv in G, that is, f^∗(uv) =

 f(u)+f(v)

2 iff(u) and f(v) are of the same parity

f(u)+f(v)+1

2 otherwise.

Then the vertex labeling f is called a mean labeling of G if its induced edge labeling f^∗ : E → {1, 2, . . . , q} is a bijection, that is, f^∗(E) = {1, 2, . . . , q}. If a graphG has a mean labeling, then we say that G is a mean graph.

Example 1.1. A mean labeling of the Petersen graph is given in Figure 1.

s

s s

s s

s s

s s s

2

4

5 3

0 9

14 13

12 15

Figure 1. A mean labeling of the Petersen graph

In [6], Sethuraman and Selvaraju introduced the concept of arbitrary super sub- division of graphs. An arbitrary super subdivision of a graphG is obtained from G by replacing every edgeuv of G by K2,m(m may vary for each edge) by identifying u and v with the vertices x and y respectively, where {x, y} is a partition of K2,m. Example 1.2.

s s

s

s

s s

s

u1 u2

vm

v2

u1=x v1 u2=y

Figure 2. The graphP2 and its arbitrary super subdivision

In [4], Ponraj and Somasundaram proved that any arbitrary super subdivision graph of the path is a mean graph. Motivated by the work in this paper, we study the meanness of another form of arbitrary path super subdivision graph ofPn. Here any arbitrary path super subdivision of a graphG is obtained from G by replacing every edgeuv of G by Pr,m(r ≥ 2 and m ≥ 2 may vary for each edge) where Pr,mis obtained fromm copies of the path of size r by identifying their corresponding end vertices.

Notation. We denote the arbitrary path super subdivision of the path Pn = u1u2. . . un by APS(Pn).

2 Meanness of arbitrary path super subdivision graph of P

Letu1, u2, . . . , un be the vertices of the pathPn. We replace each edgeusus+1(1≤ s ≤ n − 1) of PnbyPrs,ms for anyrs≥ 2 and ms≥ 2, where Prs,ms is obtained from mscopies of the path of sizersby identifying their corresponding end vertices. The resulting graph is an arbitrary path super subdivision graph ofPn.

Letu1, u2, . . . , un be the vertices of the pathPn; each edge of the pathusus+1is replaced byPrs,ms, s = 1, 2, . . . , n − 1. Let us=vⁱs0, vsⁱ1, vⁱs2, . . . , vsⁱ_rs =us+1 be the vertices of thei^thcopy of the path of sizerswherei = 1, 2, . . . , ms, s = 1, 2, . . . , n−1.

We observe that the graph APS(Pn) has

n−1

s=1(rs− 1)ms+n vertices and ⁿ⁻¹

s=1rs ms

edges.

Example 2.1. An APS(P4) when the edgeu1u2is replaced by P3,4,u2u3 is replaced byP2,5 andu3u4is replaced byP4,3 is shown in Figure 3.

t t t t t t t t t t

cc

cc

t t

t

t t

t

t t t t

t

t t

t t t

u1 v¹11 v¹12 u2 v¹21 u3 v3¹1 v¹32 v¹33 u4

v1²1 v1²2

v1³1 v1³2

v1⁴1 v1⁴2

v²21

v³21

v₂⁴₁ v⁵21

v²31 v²32 v²33

v³31 v³32 v3³3

Figure 3. An APS(P4)

Theorem 2.2. APS(Pn) is a mean graph where each edgeusus+1 ofPn is replaced byPrs,ms,ms= 2ks+ 1 for someks, s = 1, 2, . . . , n − 1 and rs≥ 2.

Proof. Let u1, u2, . . . , un be the vertices of the path Pn and let each edge of the path usus+1 (1 ≤ s ≤ n − 1) be replaced by Pr1,2k1+1, Pr2,2k2+1, . . . , Prn−1,2kn−1+1

respectively. Letus=vsⁱ0, vⁱs1, vⁱs2, . . . , vⁱs_rs =us+1 be the vertices of thei^thcopy of the path of sizers wherei = 1, 2, . . . , 2ks+ 1,s = 1, 2, . . . , n − 1. We observe that

the number of vertices of the graph APS(Pn) isⁿ⁻¹

s=1(rs− 1)ms+n and the number of edges of the graph isⁿ⁻¹

s=1rsms. We definef as follows:

f(u1) = 0

f(us) =f(us−1) + (ms−1)(rs−1), 2 ≤ s ≤ n f(vⁱs2j+1) =f(us) + 2msj + 2i − 1, 1 ≤ s ≤ n − 1

i = 1, 2, . . . , 2ks+ 1 j =

 0, 1, . . . ,^rs₂⁻¹− 1 ifrsis odd 0, 1, . . . ,^r₂^s− 1 ifrsis even

f(vⁱs2j) =

⎧⎪

⎪⎨

⎪⎪

⎩

f(us) + (6ks+ 2)

+2ms(j − 1) − 4(i − 1), 1≤ i ≤ ks+ 1 f(us) + 6ks

+2ms(j − 1) − 4(i − (ks+ 2)), ks+ 2≤ i ≤ 2ks+ 1 j =

 1, 2, . . . ,^rs₂⁻¹ ifrsis odd

1, 2, . . . ,^r₂^s− 1 ifrsis even, 1≤ s ≤ n − 1 The edge labels of APS(Pn) are given as follows:

When rs is odd or even, the labels of the edges v¹s0v¹s1, v²s0v²s1, . . . , vs^m0^sv^ms1^s are f(us) + 1, f(us) + 2, . . . , f(us) +msrespectively and for eacht, 1 ≤ t ≤ rs− 2, the labels of the edges

vs^kt^s+1vs^kt+1^s+1, v^kst^sv^kst+1^s , . . . , vs¹tvs¹t+1, v^2kst^s+1v^2kst+1^s+1, vs^2kt^svs^2kt+1^s , . . . , vs^kt^s+2v^kst+1^s+2

aref(us) +tms+ 1, f(us) +tms+ 2, . . . , f(us) + (t + 1)msrespectively.

The labels of the edges

vs^k_rs−1^s+1vs^k_rs^s+1, v^2ks_rs−1^s+1v^2ks_rs^s+1, vs^k_rs−1^s v^ks_rs^s, v^2ks_rs−1^s v^2ks_rs^s, . . . , vs²_rs−1vs²_rs, vs^k_rs−1^s+2vs^k_rs^s+2, vs¹_rs−1vs¹_rs

aref(us) + (rs− 1)ms+ 1, f(us) + (rs− 1)ms+ 2, . . . , f(us) +rsmsrespectively ifrs

is odd and the labels of the edges

vs¹_rs−1vs¹_rs, vs²_rs−1vs²_rs, . . . , vs^2k_rs−1^s+1vs^2k_rs^s+1

aref(us) + (rs− 1)ms+ 1, f(us) + (rs− 1)ms+ 2, . . . , f(us) +rsms respectively if rsis even.

Hence, in eachPrs,ms, 1≤ s ≤ n − 1, the set of parallel edges {vsⁱtvsⁱt+1 : 1≤ i ≤ ms} has distinct edge labels f(us) +tms+ 1, f(us) +tms+ 2, . . . , f(us) + (t + 1)ms, for each 0≤ t ≤ rs− 1.

Thus, the edge labels of APS(Pn) are 1, 2, 3, . . . ,ⁿ⁻¹

s=1rsmsand the edge labels are distinct. Hence APS(Pn) is a mean graph.

Example 2.3. Consider APS(P5) where the edgeu1u2 is replaced by P2,7, u2u3 is replaced byP4,3,u3u4is replaced byP5,5andu4u5is replaced byP3,7. A mean labeling of APS(P5) is shown in Figure 4.

r r r r r r r r r rr rr rr rr rr rr

0 1 14 15 22 21 26 27 40 37 50 51 52 71 72

rrr r r r

r r r

r r r

r r r r

r r r r

r r r r

r r r r

r r r r r

r r

r r r r r 3

5 7 9 11

13

17 18 23

19 20 25

29 36 39 46

31 32 41 42

33 38 43 48

35 34 45 44

64 61

54 67 56 63 58 59 60 69 62 65

Figure 4. A mean labeling of an APS(P5)

The above theorem has been proved only for odd integersms. We found it to be a hard problem to show a similar result for APS(Pn) whenmsis even andrs≥ 7. In the next theorem we considerrs∈ {2, 3, . . . , 6} whenever msis even.

Theorem 2.4. APS(P n) is a mean graph where each edge usus+1 ofPnis replaced by Prs,ms, when ms is odd; rs ≥ 2 and when ms is even; 2 ≤ rs ≤ 6, for some s = 1, 2, . . . , n − 1.

Proof. Letu1, u2, . . . , unbe the vertices of the pathPnand suppose the edgeutut+1

is replaced byPrt,mt wheremt= 2ktfor somektand we consider 2 ≤ rt≤ 6. Now we give the mean labeling of the arbitrary path super subdivision of the edgeutut+1

as follows.

Letut=vtⁱ0, vⁱt1, vⁱt2, . . . , vtⁱ_rt =ut+1be the vertices of thei^th copy of the path of lengthrtwherei = 1, 2, . . . , 2kt.

Case i. Whenrt= 2 orrt= 3 we definef as follows:

f(ut) =

⎧⎨

⎩

0, t = 1

t−1

s=1rsms, 2≤ t ≤ n

f(vtⁱ2j+1) =f(ut) + 2i − 1, i = 1, 2, . . . , 2ktandj = 0 if rt= 2, 3 f(vtⁱ2j) =

⎧⎨

⎩

f(ut) + 4kt+ 1 + 2(i − 1), 1≤ i ≤ kt

f(ut) + 2kt+ 2 + 2(i − (kt+ 1)), kt+ 1≤ i ≤ 2kt, j = 1 if rt= 3.

It can be verified that the labels of the edges aref(ut) + 1, f(ut) + 2, . . . , f(ut+1) and all the edge labels are distinct.

Case ii. Whenrt= 4 or 5 we definef as follows:

f(ut) =

⎧⎨

⎩

0, t = 1

t−1

s=1rsms, 2≤ t ≤ n

f(vtⁱ2j+1) =f(ut) + 4ktj + 2i − 1, i = 1, 2, . . . , 2ktandj = 0, 1 if rt= 4, 5

f(vtⁱ2j) =

⎧⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎩

f(ut) + 4kt

+2 + (4kt− 1)(j − 1) + 2(i − 1), 1≤ i ≤ kt

f(ut) + 2kt+ (4kt+ 2)(j − 1)

+2(i − (kt+ 1)), kt+ 1≤ i ≤ 2kt

j = 1 when rt= 4 and j = 1, 2 when rt= 5.

It is easy to check that the labels of the edges aref(ut) + 1, f(ut) + 2, . . . , f(ut+1) and all the edge labels are distinct.

Case iii. Whenrt= 6 we definef as follows:

f(ut) =

⎧⎨

⎩

0, t = 1

t−1

s=1rsms, 2≤ t ≤ n

f(vⁱt2j+1) =

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

f(ut) + 4ktj + 2i − 1, 1≤ i ≤ kt, j = 0, 1 f(ut) + (4kt+ 1)j + 2i − 1, kt+ 1≤ i ≤ 2kt, j = 0, 1 f(ut) + (4kt+ 1)j + 2(i − 1), 1≤ i ≤ kt, j = 2 f(ut) + (5kt+ 1)j − 1

+2(i − (kt+ 1)), kt+ 1≤ i ≤ 2kt, j = 2

f(vⁱt2j) =

⎧⎪

⎪⎨

⎪⎪

⎩

f(ut) + (4kt+ 2)

+(4kt− 1)(j − 1) + 2(i − 1), 1≤ i ≤ kt, j = 1, 2 f(ut) + 2kt+ (4kt+ 1)(j − 1)

+2(i − (kt+ 1)), kt+ 1≤ i ≤ 2kt, j = 1, 2.

It can be easily verified that the labels of the edges aref(ut) + 1, f(ut) + 2, . . . , f(ut+1) and all the edge labels are distinct.

For odd values ofmt, we give the labels for the vertices of arbitrary path super subdivision ofutut+1as in Theorem 2.2.

It is easy to check that the edge labels of APS(Pn) are 1, 2, 3, . . . ,ⁿ⁻¹

s=1rsmsand the edge labels are distinct. Hence APS(Pn) is a mean graph.

Example 2.5. A mean labeling of APS(P5) where the edgeu1u2 is replaced byP3,5, u2u3 is replaced byP5,4, u3u4is replaced byP2,7, u4u5 is replaced byP3,6, is shown in Figure 5.

r r r r r r r r r rr rr r rr rr

0 1 14 15 16 25 24 32 35 36 49 50 62 67

r r

r r

r r r r

r r r

r r r

r r r

r r r

rr r r r r

r r r r

r r

r r r 3 r

5 7

9 8

12 6

10 18 27 26 34

20 19 28 29

22 21 30 31

38 40 42 44 46 48

52 64

54 66

56 57

58 59

60 61

Figure 5. A mean labeling of an APS(P5)

Corollary 2.6. APS(Pn) is a mean graph where each edgeusus+1 ofPnis replaced byPrs,ms whenms= 2ksfor someks, 2 ≤ rs≤ 6, s = 1, 2, . . . , n − 1.

Proof. It follows from the labeling given in the part of the proof of Theorem 2.4 whenmtis even.

Example 2.7. Consider APS(P5) where the edge u1u2 is replaced byP2,8, u2u3 is replaced byP4,4, u3u4is replaced byP3,6andu4u5is replaced byP6,4. A mean labeling of APS(P5) is shown in Figure 6.

r r r r r r r r r rr rr rr rr rr rr

0 1 16 17 26 25 32 33 45 50 51 60 59 67 68

rrr r r r

r r r

r r r

r r r r

r r r r

r r r

r r

r r

r r

3 5 7 9 11

13

19 28 27

21 20 29

35 47 53

37 49 55

39 40

41 42

61 69

64 63

rr r

r r r

r r

r r

r r

r r

74 15

23 22 31

43 44

57 56 66 65 73

62 54

70 71

Figure 6. A mean labeling of an APS(P5)

We propose the following open problem for further research.

Problem 2.8. For what values ofrs> 6, is APS(Pn) a mean graph where each edge usus+1ofPn is replaced byPrs,ms whenmsis even?

Acknowledgements

The authors are thankful to the referees for their helpful suggestions which lead to a substantial improvement in the presentation of the paper.

References

[1] R. Balakrishnan and N. Renganathan, A Text Book on Graph Theory, Springer Verlag, 2000.

[2] A. Nagarajan and R. Vasuki, Further results on mean graphs, (preprint).

[3] R. Ponraj and S. Somasundaram, Further results on mean graphs, Proc. Sacoef- erence (2005), 443–448.

[4] R. Ponraj and S. Somasundaram, Mean labeling of graphs obtained by identifying two graphs, J. Discrete Math. Sciences Cryptography 11(2) (2008), 239–252.

[5] A. Selvam and R. Vasuki, Some results on mean graphs, Ultra Scientist of Physical Sciences 21(1) (2009), 273–284.

[6] G. Sethuraman and P. Selvaraju, Gracefulness of Arbitrary Super Subdivision of Graphs, Indian J. Pure Appl. Math. 32(7) (2001), 1059–1064.

[7] S. Somasundaram and R. Ponraj, Mean labelings of graphs, National Academy Science Letters 26 (2003), 210–213.

[8] S. Somasundaram and R. Ponraj, Some results on mean graphs, Pure Appl.

Mathematika Sciences 58 (2003), 29–35.

(Received 1 Apr 2010; revised 22 Nov 2010 and 12 May 2011)